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Magazine Chapter 6: Problem 12
\(\int \sqrt{5 r+1} d r\)
Short Answer
Expert verified
(2/15)(5r + 1)^(3/2) + C
Step by step solution
01
Identify the substitution
Letting set a new variable for easier integration. Choose the innermost function under the square root as the new variable. Set and identify u = 5r + 1.
02
Differentiate u
Now, differentiate u with respect to r so you can substitute into the integral: du/dr = 5 or du = 5 dr This allows substituting du back into the integral in terms of u.
03
Solve for dr
Next, solve for dr to complete the substitution: dr = du / 5.
04
Substitute and simplify
Substitute the expressions for u and dr into the original integral: 鈭� 鈭�(5r + 1) dr becomes 鈭� 鈭歶 (du/5) = (1/5)鈭� 鈭歶 du
05
Integrate with respect to u
Now integrate with respect to u: (1/5) 鈭� u^(1/2) du Use the power rule for integration: 鈭� u^n du = (u^(n+1))/(n+1), where n = 1/2.
06
Apply the power rule
Applying the power rule: (1/5) 鈭� u^(1/2) du = (1/5) * (u^(3/2) / (3/2)) = (1/5) * (2/3) u^(3/2) = (2/15) u^(3/2).
07
Substitute back for r
Finally, substitute back for r using u = 5r + 1: (2/15) (5r + 1)^(3/2). Don't forget to add the constant of integration, C.
08
Write the final answer
Therefore, the integral is (2/15) (5r + 1)^(3/2) + C.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
u-substitution
To solve integrals efficiently, sometimes we need to simplify the integrand. One effective method is u-substitution. This method involves substituting a part of the integrand with a new variable, usually denoted as 'u', to make the integral easier to solve. In the given exercise, identify the innermost function under the square root, which is 5r + 1. Set this equal to u, so we have u = 5r + 1.
power rule for integration
The power rule for integration is a fundamental technique used to solve integrals involving powers of variables. The rule states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), where n is not equal to -1. In the given exercise, after substitution and simplification, we use the power rule to integrate \(\frac{1}{5} \int u^{1/2} du\). This rule allows us to find the antiderivative quickly by increasing the power of u by 1 and dividing by the new power.
integration techniques
Integration techniques are various methods used to find antiderivatives of functions. Along with u-substitution and the power rule, other methods include integration by parts and partial fractions. Choosing the appropriate technique depends on the form of the integrand. In this exercise, u-substitution simplifies the problem by converting a complex integral into a simpler form that can be easily integrated using the power rule.
constant of integration
When we integrate a function, we always add a constant of integration, denoted by C. This constant accounts for the fact that indefinite integrals (antiderivatives) are not unique - they differ by a constant. In our final step, after integrating and substituting back for r, we add C to our result to ensure the solution represents the entire family of antiderivatives for the given function. Therefore, the final answer is \( \frac{2}{15} (5r+1)^{3/2} + C \).
